## Abstract

It is well known that the point group of the root lattice D_{6} admits the icosahedral group as a maximal subgroup. The generators of the icosahedral group H_{3}, its roots, and weights are determined in terms of those of D_{6}. Platonic and Archimedean solids possessing icosahedral symmetry have been obtained by projections of the sets of lattice vectors of D_{6} determined by a pair of integers (m_{1}, m_{2}) in most cases, either both even or both odd. Vertices of the Danzer’s ABCK tetrahedra are determined as the fundamental weights of H_{3}, and it is shown that the inflation of the tiles can be obtained as projections of the lattice vectors characterized by the pair of integers, which are linear combinations of the integers (m_{1}, m_{2}) with coefficients from the Fibonacci sequence. Tiling procedure both for the ABCK tetrahedral and the <ABCK> octahedral tilings in 3D space with icosahedral symmetry H_{3}, and those related transformations in 6D space with D_{6} symmetry are specified by determining the rotations and translations in 3D and the corresponding group elements in D_{6}. The tetrahedron K constitutes the fundamental region of the icosahedral group and generates the rhombic triacontahedron upon the group action. Properties of “K-polyhedron”, “B-polyhedron”, and “C-polyhedron” generated by the icosahedral group have been discussed.

Original language | English |
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Article number | 1983 |

Pages (from-to) | 1-15 |

Number of pages | 15 |

Journal | Symmetry |

Volume | 12 |

Issue number | 12 |

DOIs | |

Publication status | Published - Dec 2020 |

Externally published | Yes |

## Keywords

- Aperiodic tilings
- Coxeter–Weyl groups
- Icosahedral group
- Lattices
- Polyhedra
- Projections of polytopes
- Quasicrystals

## ASJC Scopus subject areas

- Computer Science (miscellaneous)
- Chemistry (miscellaneous)
- Mathematics(all)
- Physics and Astronomy (miscellaneous)